In Exercises 15–26, use graphs to find each set. (- 3, 0) ⋃ [- 1, 2]

Verified step by step guidance

1. Understand the notation: The symbol \( \cup \) represents the union of two sets, meaning all elements that are in either set or both.
2. Identify the sets given: The first set is \( (-3, 0) \), which is an open interval including all numbers greater than \(-3\) and less than \(0\), but not including \(-3\) or \(0\). The second set is \( [-1, 2] \), which is a closed interval including all numbers from \(-1\) to \(2\), including the endpoints \(-1\) and \(2\).
3. Visualize the intervals on a number line: Mark the points \(-3\), \(-1\), \(0\), and \(2\). For \((-3, 0)\), draw an open circle at \(-3\) and \(0\) and shade the region between them. For \([-1, 2]\), draw closed circles at \(-1\) and \(2\) and shade the region between them.
4. Determine the union \( (-3, 0) \cup [-1, 2] \): Combine all the shaded regions from both intervals. Since \([-1, 2]\) overlaps with \((-3, 0)\) between \(-1\) and \(0\), the union will cover from just greater than \(-3\) up to \(2\), including \(-1\) and \(2\) but excluding \(-3\) and \(0\) (except where included by the second set).
5. Express the union as an interval or combination of intervals: Based on the combined shading, write the union in interval notation, carefully noting which endpoints are included or excluded.

Key Concepts

Union of Sets

The union of two sets combines all elements from both sets without duplication. For intervals, this means including every number that lies in either interval. Understanding union helps in visualizing and writing the combined range of values.

Interval Notation

Interval notation is a way to represent sets of real numbers using parentheses and brackets. Parentheses ( ) indicate that an endpoint is not included (open interval), while brackets [ ] mean the endpoint is included (closed interval). This notation is essential for accurately describing sets on a number line.

Graphing Intervals on the Number Line

Graphing intervals involves marking the start and end points on a number line and shading the region between them. Open circles represent excluded endpoints, and closed circles represent included endpoints. This visual tool helps in understanding the union and intersection of sets.